3.1.45 \(\int (d x)^m (a+b \tanh ^{-1}(c x)) \, dx\) [45]

Optimal. Leaf size=72 \[ \frac {(d x)^{1+m} \left (a+b \tanh ^{-1}(c x)\right )}{d (1+m)}-\frac {b c (d x)^{2+m} \, _2F_1\left (1,\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{d^2 (1+m) (2+m)} \]

[Out]

(d*x)^(1+m)*(a+b*arctanh(c*x))/d/(1+m)-b*c*(d*x)^(2+m)*hypergeom([1, 1+1/2*m],[2+1/2*m],c^2*x^2)/d^2/(1+m)/(2+
m)

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Rubi [A]
time = 0.03, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6049, 371} \begin {gather*} \frac {(d x)^{m+1} \left (a+b \tanh ^{-1}(c x)\right )}{d (m+1)}-\frac {b c (d x)^{m+2} \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{d^2 (m+1) (m+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*(a + b*ArcTanh[c*x]),x]

[Out]

((d*x)^(1 + m)*(a + b*ArcTanh[c*x]))/(d*(1 + m)) - (b*c*(d*x)^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/
2, c^2*x^2])/(d^2*(1 + m)*(2 + m))

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 6049

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))*((d_)*(x_))^(m_), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcTan
h[c*x^n])/(d*(m + 1))), x] - Dist[b*c*(n/(d^n*(m + 1))), Int[(d*x)^(m + n)/(1 - c^2*x^(2*n)), x], x] /; FreeQ[
{a, b, c, d, m, n}, x] && IntegerQ[n] && NeQ[m, -1]

Rubi steps

\begin {align*} \int (d x)^m \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac {(d x)^{1+m} \left (a+b \tanh ^{-1}(c x)\right )}{d (1+m)}-\frac {(b c) \int \frac {(d x)^{1+m}}{1-c^2 x^2} \, dx}{d (1+m)}\\ &=\frac {(d x)^{1+m} \left (a+b \tanh ^{-1}(c x)\right )}{d (1+m)}-\frac {b c (d x)^{2+m} \, _2F_1\left (1,\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{d^2 (1+m) (2+m)}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 59, normalized size = 0.82 \begin {gather*} -\frac {x (d x)^m \left (-\left ((2+m) \left (a+b \tanh ^{-1}(c x)\right )\right )+b c x \, _2F_1\left (1,1+\frac {m}{2};2+\frac {m}{2};c^2 x^2\right )\right )}{(1+m) (2+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m*(a + b*ArcTanh[c*x]),x]

[Out]

-((x*(d*x)^m*(-((2 + m)*(a + b*ArcTanh[c*x])) + b*c*x*Hypergeometric2F1[1, 1 + m/2, 2 + m/2, c^2*x^2]))/((1 +
m)*(2 + m)))

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Maple [F]
time = 0.25, size = 0, normalized size = 0.00 \[\int \left (d x \right )^{m} \left (a +b \arctanh \left (c x \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a+b*arctanh(c*x)),x)

[Out]

int((d*x)^m*(a+b*arctanh(c*x)),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+b*arctanh(c*x)),x, algorithm="maxima")

[Out]

1/2*(2*c*d^m*integrate(x*x^m/(c^2*(m + 1)*x^2 - m - 1), x) + (d^m*x*x^m*log(c*x + 1) - d^m*x*x^m*log(-c*x + 1)
)/(m + 1))*b + (d*x)^(m + 1)*a/(d*(m + 1))

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+b*arctanh(c*x)),x, algorithm="fricas")

[Out]

integral((b*arctanh(c*x) + a)*(d*x)^m, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d x\right )^{m} \left (a + b \operatorname {atanh}{\left (c x \right )}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*(a+b*atanh(c*x)),x)

[Out]

Integral((d*x)**m*(a + b*atanh(c*x)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+b*arctanh(c*x)),x, algorithm="giac")

[Out]

integrate((b*arctanh(c*x) + a)*(d*x)^m, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,{\left (d\,x\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x))*(d*x)^m,x)

[Out]

int((a + b*atanh(c*x))*(d*x)^m, x)

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